Inverse Limit Shape Problem for Multiplicative Ensembles of Convex Lattice Polygonal Lines

Convex polygonal lines with vertices in Z + 2 and endpoints at 0 = ( 0 , 0 ) and n = ( n 1 , n 2 ) → ∞ , such that n 2 / n 1 → c ∈ ( 0 , ∞ ) , under the scaling n 1 − 1 , have limit shape γ * with respect to the uniform distribution, identified as the parabola arc c ( 1 − x 1 ) + x 2 = c . This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of γ * , we demonstrate that, for any strictly convex C 3 -smooth arc γ ⊂ R + 2 started at the origin and with the slope at each point not exceeding 90 ∘ , there is a sequence of multiplicative probability measures P n γ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape.